Find subgradient of $f(x)=max{f_1(x), f_2(x)}$
$begingroup$
Let $f(x)=max{f_1(x), f_2(x)}$ where $f_1$ and $f_2$ are differentiable convex functions defined on $R^n$. Let $x'$ be such that $f(x')=f_1(x')=f_2(x')$.
Show that $g$ is a subgradient of f at $x'$ if and only if $$g=lambda nabla f_1(x')+(1-lambda) nabla f_2(x')$$ where $0leq lambda leq1.$
The "if" part was pretty straight-forward, but as often I tend to struggle with the "only if" part. I thought of assuming that there exists a subgradient $g$ such that $g notin conv(nabla f_1(x'), nabla f_2(x'))$ and using the theorem of hyperplane seperation, but so far I've had no luck.
In general I tend to struggle with proving that every subgradient has a particular form.
convex-optimization
edited Dec 11 '18 at 17:46
user1101010
7391730
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
$endgroup$
add a comment |
$begingroup$
Let $f(x)=max{f_1(x), f_2(x)}$ where $f_1$ and $f_2$ are differentiable convex functions defined on $R^n$. Let $x'$ be such that $f(x')=f_1(x')=f_2(x')$.
Show that $g$ is a subgradient of f at $x'$ if and only if $$g=lambda nabla f_1(x')+(1-lambda) nabla f_2(x')$$ where $0leq lambda leq1.$
The "if" part was pretty straight-forward, but as often I tend to struggle with the "only if" part. I thought of assuming that there exists a subgradient $g$ such that $g notin conv(nabla f_1(x'), nabla f_2(x'))$ and using the theorem of hyperplane seperation, but so far I've had no luck.
In general I tend to struggle with proving that every subgradient has a particular form.
convex-optimization
edited Dec 11 '18 at 17:46
user1101010
7391730
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
$endgroup$
add a comment |
$begingroup$
Let $f(x)=max{f_1(x), f_2(x)}$ where $f_1$ and $f_2$ are differentiable convex functions defined on $R^n$. Let $x'$ be such that $f(x')=f_1(x')=f_2(x')$.
Show that $g$ is a subgradient of f at $x'$ if and only if $$g=lambda nabla f_1(x')+(1-lambda) nabla f_2(x')$$ where $0leq lambda leq1.$
The "if" part was pretty straight-forward, but as often I tend to struggle with the "only if" part. I thought of assuming that there exists a subgradient $g$ such that $g notin conv(nabla f_1(x'), nabla f_2(x'))$ and using the theorem of hyperplane seperation, but so far I've had no luck.
In general I tend to struggle with proving that every subgradient has a particular form.
convex-optimization
edited Dec 11 '18 at 17:46
user1101010
7391730
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
$endgroup$
Let $f(x)=max{f_1(x), f_2(x)}$ where $f_1$ and $f_2$ are differentiable convex functions defined on $R^n$. Let $x'$ be such that $f(x')=f_1(x')=f_2(x')$.
Show that $g$ is a subgradient of f at $x'$ if and only if $$g=lambda nabla f_1(x')+(1-lambda) nabla f_2(x')$$ where $0leq lambda leq1.$
The "if" part was pretty straight-forward, but as often I tend to struggle with the "only if" part. I thought of assuming that there exists a subgradient $g$ such that $g notin conv(nabla f_1(x'), nabla f_2(x'))$ and using the theorem of hyperplane seperation, but so far I've had no luck.
In general I tend to struggle with proving that every subgradient has a particular form.
convex-optimization
convex-optimization
edited Dec 11 '18 at 17:46
user1101010
7391730
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
edited Dec 11 '18 at 17:46
user1101010
7391730
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
edited Dec 11 '18 at 17:46
user1101010
7391730
edited Dec 11 '18 at 17:46
user1101010
7391730
edited Dec 11 '18 at 17:46
user1101010
7391730
7391730
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
asked Dec 11 '18 at 15:59
AGunnarAGunnar
204
204
add a comment |
add a comment |
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